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            <h2 class="toc-title">目录</h2>
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        </div><article class="page single"><h1 class="single-title animated flipInX">如何理解系统的可观测性</h1><div class="post-meta">
            <div class="post-meta-line"><span class="post-author"><a href="/" title="Author" rel=" author" class="author"><i class="fas fa-user-circle fa-fw"></i>辛载阳</a></span>&nbsp;<span class="post-category">收录于 <a href="/categories/%E7%9F%A5%E8%AF%86/"><i class="far fa-folder fa-fw"></i>知识</a></span></div>
            <div class="post-meta-line"><i class="far fa-calendar-alt fa-fw"></i>&nbsp;<time datetime="2019-05-05">2019-05-05</time>&nbsp;<i class="fas fa-pencil-alt fa-fw"></i>&nbsp;约 761 字&nbsp;
                <i class="far fa-clock fa-fw"></i>&nbsp;预计阅读 2 分钟&nbsp;</div>
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            </div><div class="content" id="content"><p>系统的可观测性的定义为：</p>
<p>对于系统 <img
        class="lazyload"
        src="/svg/loading.min.svg"
        data-src="https://www.zhihu.com/equation?tex=%5Cbegin%7Baligned%7D&#43;%5Cdot%7Bx%7D%28t%29&#43;%26%3DA%28t%29&#43;x%28t%29&#43;%5C%5C&#43;y%28t%29&#43;%26%3DC%28t%29&#43;x%28t%29&#43;%5Cend%7Baligned%7D"
        data-srcset="https://www.zhihu.com/equation?tex=%5Cbegin%7Baligned%7D&#43;%5Cdot%7Bx%7D%28t%29&#43;%26%3DA%28t%29&#43;x%28t%29&#43;%5C%5C&#43;y%28t%29&#43;%26%3DC%28t%29&#43;x%28t%29&#43;%5Cend%7Baligned%7D, https://www.zhihu.com/equation?tex=%5Cbegin%7Baligned%7D&#43;%5Cdot%7Bx%7D%28t%29&#43;%26%3DA%28t%29&#43;x%28t%29&#43;%5C%5C&#43;y%28t%29&#43;%26%3DC%28t%29&#43;x%28t%29&#43;%5Cend%7Baligned%7D 1.5x, https://www.zhihu.com/equation?tex=%5Cbegin%7Baligned%7D&#43;%5Cdot%7Bx%7D%28t%29&#43;%26%3DA%28t%29&#43;x%28t%29&#43;%5C%5C&#43;y%28t%29&#43;%26%3DC%28t%29&#43;x%28t%29&#43;%5Cend%7Baligned%7D 2x"
        data-sizes="auto"
        alt="https://www.zhihu.com/equation?tex=%5Cbegin%7Baligned%7D&#43;%5Cdot%7Bx%7D%28t%29&#43;%26%3DA%28t%29&#43;x%28t%29&#43;%5C%5C&#43;y%28t%29&#43;%26%3DC%28t%29&#43;x%28t%29&#43;%5Cend%7Baligned%7D"
        title="[公式]" /> 在有限的时间区间里 <img
        class="lazyload"
        src="/svg/loading.min.svg"
        data-src="https://www.zhihu.com/equation?tex=%5Bt_0%2Ct_%7B%5Calpha%7D%5D"
        data-srcset="https://www.zhihu.com/equation?tex=%5Bt_0%2Ct_%7B%5Calpha%7D%5D, https://www.zhihu.com/equation?tex=%5Bt_0%2Ct_%7B%5Calpha%7D%5D 1.5x, https://www.zhihu.com/equation?tex=%5Bt_0%2Ct_%7B%5Calpha%7D%5D 2x"
        data-sizes="auto"
        alt="https://www.zhihu.com/equation?tex=%5Bt_0%2Ct_%7B%5Calpha%7D%5D"
        title="[公式]" /> 内，对应初态 <img
        class="lazyload"
        src="/svg/loading.min.svg"
        data-src="https://www.zhihu.com/equation?tex=x%28t_0%29&#43;%3D&#43;%5Cbar%7Bx%7D"
        data-srcset="https://www.zhihu.com/equation?tex=x%28t_0%29&#43;%3D&#43;%5Cbar%7Bx%7D, https://www.zhihu.com/equation?tex=x%28t_0%29&#43;%3D&#43;%5Cbar%7Bx%7D 1.5x, https://www.zhihu.com/equation?tex=x%28t_0%29&#43;%3D&#43;%5Cbar%7Bx%7D 2x"
        data-sizes="auto"
        alt="https://www.zhihu.com/equation?tex=x%28t_0%29&#43;%3D&#43;%5Cbar%7Bx%7D"
        title="[公式]" /> ，有</p>
<p><img
        class="lazyload"
        src="/svg/loading.min.svg"
        data-src="https://www.zhihu.com/equation?tex=y%28t%29&#43;%5Cequiv&#43;0%2C&#43;%5Cquad&#43;t&#43;%5Cin%5Cleft%5Bt_%7B0%7D%2C&#43;t_%7B%5Calpha%7D%5Cright%5D"
        data-srcset="https://www.zhihu.com/equation?tex=y%28t%29&#43;%5Cequiv&#43;0%2C&#43;%5Cquad&#43;t&#43;%5Cin%5Cleft%5Bt_%7B0%7D%2C&#43;t_%7B%5Calpha%7D%5Cright%5D, https://www.zhihu.com/equation?tex=y%28t%29&#43;%5Cequiv&#43;0%2C&#43;%5Cquad&#43;t&#43;%5Cin%5Cleft%5Bt_%7B0%7D%2C&#43;t_%7B%5Calpha%7D%5Cright%5D 1.5x, https://www.zhihu.com/equation?tex=y%28t%29&#43;%5Cequiv&#43;0%2C&#43;%5Cquad&#43;t&#43;%5Cin%5Cleft%5Bt_%7B0%7D%2C&#43;t_%7B%5Calpha%7D%5Cright%5D 2x"
        data-sizes="auto"
        alt="https://www.zhihu.com/equation?tex=y%28t%29&#43;%5Cequiv&#43;0%2C&#43;%5Cquad&#43;t&#43;%5Cin%5Cleft%5Bt_%7B0%7D%2C&#43;t_%7B%5Calpha%7D%5Cright%5D"
        title="[公式]" /> 则称为不可观测状态，如果不存在这样的不可观测状态，那么系统完全可观测。</p>
<p>这个定义说的非常的抽象，但是可观测性这个概念最重要的是其在 状态观测器(state observer)的设计中发挥了关键作用，下面我就从这个角度进行一些对于状态可观性的介绍。</p>
<hr>
<p>其实，状态可观性的概念是出自于卡尔曼，而著名的Kalman filter也实质上是state estimator for stochastic system[<a href="https://www.zhihu.com/people/feiyuxiaoTHU/answers/by_votes?page=2#ref_1" target="_blank" rel="noopener noreffer">1]</a>。</p>
<p>对于系统进行控制，首先我们需要确定要进行的控制的类型，一般而言，控制分为基于状态变量的控制和基于输出的控制，这其中最好的当然是状态控制，但是我们往往只有输出和输入，而难以直接得到系统的状态变量，这个时候就需要从前面所说的输出和输入还原出系统的状态变量，这就是所谓的状态观测[<a href="https://www.zhihu.com/people/feiyuxiaoTHU/answers/by_votes?page=2#ref_2" target="_blank" rel="noopener noreffer">2]</a>。</p>
<p><img
        class="lazyload"
        src="/svg/loading.min.svg"
        data-src="https://pic4.zhimg.com/80/v2-060801e2b7592a23a3828b7046713790_1440w.jpg"
        data-srcset="https://pic4.zhimg.com/80/v2-060801e2b7592a23a3828b7046713790_1440w.jpg, https://pic4.zhimg.com/80/v2-060801e2b7592a23a3828b7046713790_1440w.jpg 1.5x, https://pic4.zhimg.com/80/v2-060801e2b7592a23a3828b7046713790_1440w.jpg 2x"
        data-sizes="auto"
        alt="https://pic4.zhimg.com/80/v2-060801e2b7592a23a3828b7046713790_1440w.jpg"
        title="img" />状态反馈和输出反馈</p>
<p><img
        class="lazyload"
        src="/svg/loading.min.svg"
        data-src="https://pic3.zhimg.com/80/v2-3266d61a72e80409f698d100d13f1805_1440w.jpg"
        data-srcset="https://pic3.zhimg.com/80/v2-3266d61a72e80409f698d100d13f1805_1440w.jpg, https://pic3.zhimg.com/80/v2-3266d61a72e80409f698d100d13f1805_1440w.jpg 1.5x, https://pic3.zhimg.com/80/v2-3266d61a72e80409f698d100d13f1805_1440w.jpg 2x"
        data-sizes="auto"
        alt="https://pic3.zhimg.com/80/v2-3266d61a72e80409f698d100d13f1805_1440w.jpg"
        title="img" />典型状态观测器结构</p>
<hr>
<p><strong>一言以蔽之，系统状态的可观测性是进行状态观测器设计的前提，也就是说，无论是用什么方法，只有当系统完全可观时，才有可能从状态观测器的设计得出系统的状态的观测。</strong></p>
<p>以Full-Order Observers为例[<a href="https://www.zhihu.com/people/feiyuxiaoTHU/answers/by_votes?page=2#ref_3" target="_blank" rel="noopener noreffer">3]</a>，其结构如下:</p>
<p><img
        class="lazyload"
        src="/svg/loading.min.svg"
        data-src="https://pic3.zhimg.com/80/v2-930737553d9c84c4b6ccccbb681cedc3_1440w.jpg"
        data-srcset="https://pic3.zhimg.com/80/v2-930737553d9c84c4b6ccccbb681cedc3_1440w.jpg, https://pic3.zhimg.com/80/v2-930737553d9c84c4b6ccccbb681cedc3_1440w.jpg 1.5x, https://pic3.zhimg.com/80/v2-930737553d9c84c4b6ccccbb681cedc3_1440w.jpg 2x"
        data-sizes="auto"
        alt="https://pic3.zhimg.com/80/v2-930737553d9c84c4b6ccccbb681cedc3_1440w.jpg"
        title="img" /></p>
<p>对于观测系统有： <img
        class="lazyload"
        src="/svg/loading.min.svg"
        data-src="https://www.zhihu.com/equation?tex=%5Cdot%7B%5Chat%7Bx%7D%7D&#43;%3D&#43;A&#43;%5Chat%7Bx%7D&#43;%3D&#43;Bu&#43;%5Cquad&#43;%5Chat%7By%7D&#43;%3D&#43;&#43;C&#43;%5Chat%7Bx%7D"
        data-srcset="https://www.zhihu.com/equation?tex=%5Cdot%7B%5Chat%7Bx%7D%7D&#43;%3D&#43;A&#43;%5Chat%7Bx%7D&#43;%3D&#43;Bu&#43;%5Cquad&#43;%5Chat%7By%7D&#43;%3D&#43;&#43;C&#43;%5Chat%7Bx%7D, https://www.zhihu.com/equation?tex=%5Cdot%7B%5Chat%7Bx%7D%7D&#43;%3D&#43;A&#43;%5Chat%7Bx%7D&#43;%3D&#43;Bu&#43;%5Cquad&#43;%5Chat%7By%7D&#43;%3D&#43;&#43;C&#43;%5Chat%7Bx%7D 1.5x, https://www.zhihu.com/equation?tex=%5Cdot%7B%5Chat%7Bx%7D%7D&#43;%3D&#43;A&#43;%5Chat%7Bx%7D&#43;%3D&#43;Bu&#43;%5Cquad&#43;%5Chat%7By%7D&#43;%3D&#43;&#43;C&#43;%5Chat%7Bx%7D 2x"
        data-sizes="auto"
        alt="https://www.zhihu.com/equation?tex=%5Cdot%7B%5Chat%7Bx%7D%7D&#43;%3D&#43;A&#43;%5Chat%7Bx%7D&#43;%3D&#43;Bu&#43;%5Cquad&#43;%5Chat%7By%7D&#43;%3D&#43;&#43;C&#43;%5Chat%7Bx%7D"
        title="[公式]" /></p>
<p>跟踪误差为： <img
        class="lazyload"
        src="/svg/loading.min.svg"
        data-src="https://www.zhihu.com/equation?tex=%5Cwidetilde%7By%7D%3Dy-%5Chat%7By%7D%3DC&#43;x-C&#43;%5Chat%7Bx%7D"
        data-srcset="https://www.zhihu.com/equation?tex=%5Cwidetilde%7By%7D%3Dy-%5Chat%7By%7D%3DC&#43;x-C&#43;%5Chat%7Bx%7D, https://www.zhihu.com/equation?tex=%5Cwidetilde%7By%7D%3Dy-%5Chat%7By%7D%3DC&#43;x-C&#43;%5Chat%7Bx%7D 1.5x, https://www.zhihu.com/equation?tex=%5Cwidetilde%7By%7D%3Dy-%5Chat%7By%7D%3DC&#43;x-C&#43;%5Chat%7Bx%7D 2x"
        data-sizes="auto"
        alt="https://www.zhihu.com/equation?tex=%5Cwidetilde%7By%7D%3Dy-%5Chat%7By%7D%3DC&#43;x-C&#43;%5Chat%7Bx%7D"
        title="[公式]" /></p>
<p>即： <img
        class="lazyload"
        src="/svg/loading.min.svg"
        data-src="https://www.zhihu.com/equation?tex=%5Cdot%7B%5Chat%7Bx%7D%7D%3DA&#43;%5Chat%7Bx%7D%2BB&#43;u%2BL%28y-%5Chat%7By%7D%29"
        data-srcset="https://www.zhihu.com/equation?tex=%5Cdot%7B%5Chat%7Bx%7D%7D%3DA&#43;%5Chat%7Bx%7D%2BB&#43;u%2BL%28y-%5Chat%7By%7D%29, https://www.zhihu.com/equation?tex=%5Cdot%7B%5Chat%7Bx%7D%7D%3DA&#43;%5Chat%7Bx%7D%2BB&#43;u%2BL%28y-%5Chat%7By%7D%29 1.5x, https://www.zhihu.com/equation?tex=%5Cdot%7B%5Chat%7Bx%7D%7D%3DA&#43;%5Chat%7Bx%7D%2BB&#43;u%2BL%28y-%5Chat%7By%7D%29 2x"
        data-sizes="auto"
        alt="https://www.zhihu.com/equation?tex=%5Cdot%7B%5Chat%7Bx%7D%7D%3DA&#43;%5Chat%7Bx%7D%2BB&#43;u%2BL%28y-%5Chat%7By%7D%29"
        title="[公式]" /> 和原始系统做差有：</p>
<p><img
        class="lazyload"
        src="/svg/loading.min.svg"
        data-src="https://www.zhihu.com/equation?tex=%5Cbegin%7Baligned%7D&#43;%5Cdot%7B%5Cboldsymbol%7Bx%7D%7D-%5Cdot%7B%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%7D&#43;%26%3D%5Cboldsymbol%7BA%7D%28%5Cboldsymbol%7Bx%7D-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%29-%5Cboldsymbol%7BL%7D%28%5Cboldsymbol%7By%7D-%5Chat%7B%5Cboldsymbol%7By%7D%7D%29&#43;%5C%5C&#43;%26%3D%5Cboldsymbol%7BA%7D%28%5Cboldsymbol%7Bx%7D-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%29-%5Cboldsymbol%7BL%7D&#43;%5Cboldsymbol%7BC%7D%28%5Cboldsymbol%7Bx%7D-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%29&#43;%5C%5C&#43;%26%3D%28%5Cboldsymbol%7BA%7D-%5Cboldsymbol%7BL%7D&#43;%5Cboldsymbol%7BC%7D%29%28%5Cboldsymbol%7Bx%7D-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%29&#43;%5Cend%7Baligned%7D"
        data-srcset="https://www.zhihu.com/equation?tex=%5Cbegin%7Baligned%7D&#43;%5Cdot%7B%5Cboldsymbol%7Bx%7D%7D-%5Cdot%7B%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%7D&#43;%26%3D%5Cboldsymbol%7BA%7D%28%5Cboldsymbol%7Bx%7D-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%29-%5Cboldsymbol%7BL%7D%28%5Cboldsymbol%7By%7D-%5Chat%7B%5Cboldsymbol%7By%7D%7D%29&#43;%5C%5C&#43;%26%3D%5Cboldsymbol%7BA%7D%28%5Cboldsymbol%7Bx%7D-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%29-%5Cboldsymbol%7BL%7D&#43;%5Cboldsymbol%7BC%7D%28%5Cboldsymbol%7Bx%7D-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%29&#43;%5C%5C&#43;%26%3D%28%5Cboldsymbol%7BA%7D-%5Cboldsymbol%7BL%7D&#43;%5Cboldsymbol%7BC%7D%29%28%5Cboldsymbol%7Bx%7D-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%29&#43;%5Cend%7Baligned%7D, https://www.zhihu.com/equation?tex=%5Cbegin%7Baligned%7D&#43;%5Cdot%7B%5Cboldsymbol%7Bx%7D%7D-%5Cdot%7B%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%7D&#43;%26%3D%5Cboldsymbol%7BA%7D%28%5Cboldsymbol%7Bx%7D-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%29-%5Cboldsymbol%7BL%7D%28%5Cboldsymbol%7By%7D-%5Chat%7B%5Cboldsymbol%7By%7D%7D%29&#43;%5C%5C&#43;%26%3D%5Cboldsymbol%7BA%7D%28%5Cboldsymbol%7Bx%7D-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%29-%5Cboldsymbol%7BL%7D&#43;%5Cboldsymbol%7BC%7D%28%5Cboldsymbol%7Bx%7D-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%29&#43;%5C%5C&#43;%26%3D%28%5Cboldsymbol%7BA%7D-%5Cboldsymbol%7BL%7D&#43;%5Cboldsymbol%7BC%7D%29%28%5Cboldsymbol%7Bx%7D-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%29&#43;%5Cend%7Baligned%7D 1.5x, https://www.zhihu.com/equation?tex=%5Cbegin%7Baligned%7D&#43;%5Cdot%7B%5Cboldsymbol%7Bx%7D%7D-%5Cdot%7B%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%7D&#43;%26%3D%5Cboldsymbol%7BA%7D%28%5Cboldsymbol%7Bx%7D-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%29-%5Cboldsymbol%7BL%7D%28%5Cboldsymbol%7By%7D-%5Chat%7B%5Cboldsymbol%7By%7D%7D%29&#43;%5C%5C&#43;%26%3D%5Cboldsymbol%7BA%7D%28%5Cboldsymbol%7Bx%7D-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%29-%5Cboldsymbol%7BL%7D&#43;%5Cboldsymbol%7BC%7D%28%5Cboldsymbol%7Bx%7D-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%29&#43;%5C%5C&#43;%26%3D%28%5Cboldsymbol%7BA%7D-%5Cboldsymbol%7BL%7D&#43;%5Cboldsymbol%7BC%7D%29%28%5Cboldsymbol%7Bx%7D-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%29&#43;%5Cend%7Baligned%7D 2x"
        data-sizes="auto"
        alt="https://www.zhihu.com/equation?tex=%5Cbegin%7Baligned%7D&#43;%5Cdot%7B%5Cboldsymbol%7Bx%7D%7D-%5Cdot%7B%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%7D&#43;%26%3D%5Cboldsymbol%7BA%7D%28%5Cboldsymbol%7Bx%7D-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%29-%5Cboldsymbol%7BL%7D%28%5Cboldsymbol%7By%7D-%5Chat%7B%5Cboldsymbol%7By%7D%7D%29&#43;%5C%5C&#43;%26%3D%5Cboldsymbol%7BA%7D%28%5Cboldsymbol%7Bx%7D-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%29-%5Cboldsymbol%7BL%7D&#43;%5Cboldsymbol%7BC%7D%28%5Cboldsymbol%7Bx%7D-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%29&#43;%5C%5C&#43;%26%3D%28%5Cboldsymbol%7BA%7D-%5Cboldsymbol%7BL%7D&#43;%5Cboldsymbol%7BC%7D%29%28%5Cboldsymbol%7Bx%7D-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%29&#43;%5Cend%7Baligned%7D"
        title="[公式]" /></p>
<p>跟踪误差可以求得： <img
        class="lazyload"
        src="/svg/loading.min.svg"
        data-src="https://www.zhihu.com/equation?tex=%5Cboldsymbol%7Bx%7D%28t%29-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%28t%29%3D%5Cmathrm%7Be%7D%5E%7B%28%5Cboldsymbol%7BA%7D-%5Cboldsymbol%7BL%7D&#43;%5Cboldsymbol%7BC%7D%29%5Cleft%28t-t_%7B0%7D%5Cright%29%7D%5Cleft%5B%5Cboldsymbol%7Bx%7D%5Cleft%28t_%7B0%7D%5Cright%29-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%5Cleft%28t_%7B0%7D%5Cright%29%5Cright%5D"
        data-srcset="https://www.zhihu.com/equation?tex=%5Cboldsymbol%7Bx%7D%28t%29-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%28t%29%3D%5Cmathrm%7Be%7D%5E%7B%28%5Cboldsymbol%7BA%7D-%5Cboldsymbol%7BL%7D&#43;%5Cboldsymbol%7BC%7D%29%5Cleft%28t-t_%7B0%7D%5Cright%29%7D%5Cleft%5B%5Cboldsymbol%7Bx%7D%5Cleft%28t_%7B0%7D%5Cright%29-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%5Cleft%28t_%7B0%7D%5Cright%29%5Cright%5D, https://www.zhihu.com/equation?tex=%5Cboldsymbol%7Bx%7D%28t%29-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%28t%29%3D%5Cmathrm%7Be%7D%5E%7B%28%5Cboldsymbol%7BA%7D-%5Cboldsymbol%7BL%7D&#43;%5Cboldsymbol%7BC%7D%29%5Cleft%28t-t_%7B0%7D%5Cright%29%7D%5Cleft%5B%5Cboldsymbol%7Bx%7D%5Cleft%28t_%7B0%7D%5Cright%29-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%5Cleft%28t_%7B0%7D%5Cright%29%5Cright%5D 1.5x, https://www.zhihu.com/equation?tex=%5Cboldsymbol%7Bx%7D%28t%29-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%28t%29%3D%5Cmathrm%7Be%7D%5E%7B%28%5Cboldsymbol%7BA%7D-%5Cboldsymbol%7BL%7D&#43;%5Cboldsymbol%7BC%7D%29%5Cleft%28t-t_%7B0%7D%5Cright%29%7D%5Cleft%5B%5Cboldsymbol%7Bx%7D%5Cleft%28t_%7B0%7D%5Cright%29-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%5Cleft%28t_%7B0%7D%5Cright%29%5Cright%5D 2x"
        data-sizes="auto"
        alt="https://www.zhihu.com/equation?tex=%5Cboldsymbol%7Bx%7D%28t%29-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%28t%29%3D%5Cmathrm%7Be%7D%5E%7B%28%5Cboldsymbol%7BA%7D-%5Cboldsymbol%7BL%7D&#43;%5Cboldsymbol%7BC%7D%29%5Cleft%28t-t_%7B0%7D%5Cright%29%7D%5Cleft%5B%5Cboldsymbol%7Bx%7D%5Cleft%28t_%7B0%7D%5Cright%29-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%5Cleft%28t_%7B0%7D%5Cright%29%5Cright%5D"
        title="[公式]" /></p>
<p>写成矩阵形式：</p>
<p><img
        class="lazyload"
        src="/svg/loading.min.svg"
        data-src="https://www.zhihu.com/equation?tex=%5Cleft%5B&#43;%5Cbegin%7Barray%7D%7Bc%7D%7B%5Cdot%7B%5Cboldsymbol%7Bx%7D%7D%7D&#43;%5C%5C&#43;%7B%5Cdot%7B%5Cboldsymbol%7Bx%7D%7D-%5Cdot%7B%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%7D%7D%5Cend%7Barray%7D%5Cright%5D%3D%5Cleft%5B&#43;%5Cbegin%7Barray%7D%7Bcc%7D%7B%5Cboldsymbol%7BA%7D%7D&#43;%26&#43;%7B0%7D&#43;%5C%5C&#43;%7B0%7D&#43;%26&#43;%7B%5Cboldsymbol%7BA%7D-%5Cboldsymbol%7BL&#43;C%7D%7D%5Cend%7Barray%7D%5Cright%5D&#43;%5Cleft%5B&#43;%5Cbegin%7Barray%7D%7Bc%7D%7B%5Cboldsymbol%7Bx%7D%7D&#43;%5C%5C&#43;%7B%5Cboldsymbol%7Bx%7D-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%7D%5Cend%7Barray%7D%5Cright%5D%2B%5Cleft%5B&#43;%5Cbegin%7Barray%7D%7Bl%7D%7B%5Cboldsymbol%7BB%7D%7D&#43;%5C%5C&#43;%7B0%7D%5Cend%7Barray%7D%5Cright%5D&#43;%5Cboldsymbol%7Bu%7D"
        data-srcset="https://www.zhihu.com/equation?tex=%5Cleft%5B&#43;%5Cbegin%7Barray%7D%7Bc%7D%7B%5Cdot%7B%5Cboldsymbol%7Bx%7D%7D%7D&#43;%5C%5C&#43;%7B%5Cdot%7B%5Cboldsymbol%7Bx%7D%7D-%5Cdot%7B%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%7D%7D%5Cend%7Barray%7D%5Cright%5D%3D%5Cleft%5B&#43;%5Cbegin%7Barray%7D%7Bcc%7D%7B%5Cboldsymbol%7BA%7D%7D&#43;%26&#43;%7B0%7D&#43;%5C%5C&#43;%7B0%7D&#43;%26&#43;%7B%5Cboldsymbol%7BA%7D-%5Cboldsymbol%7BL&#43;C%7D%7D%5Cend%7Barray%7D%5Cright%5D&#43;%5Cleft%5B&#43;%5Cbegin%7Barray%7D%7Bc%7D%7B%5Cboldsymbol%7Bx%7D%7D&#43;%5C%5C&#43;%7B%5Cboldsymbol%7Bx%7D-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%7D%5Cend%7Barray%7D%5Cright%5D%2B%5Cleft%5B&#43;%5Cbegin%7Barray%7D%7Bl%7D%7B%5Cboldsymbol%7BB%7D%7D&#43;%5C%5C&#43;%7B0%7D%5Cend%7Barray%7D%5Cright%5D&#43;%5Cboldsymbol%7Bu%7D, https://www.zhihu.com/equation?tex=%5Cleft%5B&#43;%5Cbegin%7Barray%7D%7Bc%7D%7B%5Cdot%7B%5Cboldsymbol%7Bx%7D%7D%7D&#43;%5C%5C&#43;%7B%5Cdot%7B%5Cboldsymbol%7Bx%7D%7D-%5Cdot%7B%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%7D%7D%5Cend%7Barray%7D%5Cright%5D%3D%5Cleft%5B&#43;%5Cbegin%7Barray%7D%7Bcc%7D%7B%5Cboldsymbol%7BA%7D%7D&#43;%26&#43;%7B0%7D&#43;%5C%5C&#43;%7B0%7D&#43;%26&#43;%7B%5Cboldsymbol%7BA%7D-%5Cboldsymbol%7BL&#43;C%7D%7D%5Cend%7Barray%7D%5Cright%5D&#43;%5Cleft%5B&#43;%5Cbegin%7Barray%7D%7Bc%7D%7B%5Cboldsymbol%7Bx%7D%7D&#43;%5C%5C&#43;%7B%5Cboldsymbol%7Bx%7D-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%7D%5Cend%7Barray%7D%5Cright%5D%2B%5Cleft%5B&#43;%5Cbegin%7Barray%7D%7Bl%7D%7B%5Cboldsymbol%7BB%7D%7D&#43;%5C%5C&#43;%7B0%7D%5Cend%7Barray%7D%5Cright%5D&#43;%5Cboldsymbol%7Bu%7D 1.5x, https://www.zhihu.com/equation?tex=%5Cleft%5B&#43;%5Cbegin%7Barray%7D%7Bc%7D%7B%5Cdot%7B%5Cboldsymbol%7Bx%7D%7D%7D&#43;%5C%5C&#43;%7B%5Cdot%7B%5Cboldsymbol%7Bx%7D%7D-%5Cdot%7B%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%7D%7D%5Cend%7Barray%7D%5Cright%5D%3D%5Cleft%5B&#43;%5Cbegin%7Barray%7D%7Bcc%7D%7B%5Cboldsymbol%7BA%7D%7D&#43;%26&#43;%7B0%7D&#43;%5C%5C&#43;%7B0%7D&#43;%26&#43;%7B%5Cboldsymbol%7BA%7D-%5Cboldsymbol%7BL&#43;C%7D%7D%5Cend%7Barray%7D%5Cright%5D&#43;%5Cleft%5B&#43;%5Cbegin%7Barray%7D%7Bc%7D%7B%5Cboldsymbol%7Bx%7D%7D&#43;%5C%5C&#43;%7B%5Cboldsymbol%7Bx%7D-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%7D%5Cend%7Barray%7D%5Cright%5D%2B%5Cleft%5B&#43;%5Cbegin%7Barray%7D%7Bl%7D%7B%5Cboldsymbol%7BB%7D%7D&#43;%5C%5C&#43;%7B0%7D%5Cend%7Barray%7D%5Cright%5D&#43;%5Cboldsymbol%7Bu%7D 2x"
        data-sizes="auto"
        alt="https://www.zhihu.com/equation?tex=%5Cleft%5B&#43;%5Cbegin%7Barray%7D%7Bc%7D%7B%5Cdot%7B%5Cboldsymbol%7Bx%7D%7D%7D&#43;%5C%5C&#43;%7B%5Cdot%7B%5Cboldsymbol%7Bx%7D%7D-%5Cdot%7B%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%7D%7D%5Cend%7Barray%7D%5Cright%5D%3D%5Cleft%5B&#43;%5Cbegin%7Barray%7D%7Bcc%7D%7B%5Cboldsymbol%7BA%7D%7D&#43;%26&#43;%7B0%7D&#43;%5C%5C&#43;%7B0%7D&#43;%26&#43;%7B%5Cboldsymbol%7BA%7D-%5Cboldsymbol%7BL&#43;C%7D%7D%5Cend%7Barray%7D%5Cright%5D&#43;%5Cleft%5B&#43;%5Cbegin%7Barray%7D%7Bc%7D%7B%5Cboldsymbol%7Bx%7D%7D&#43;%5C%5C&#43;%7B%5Cboldsymbol%7Bx%7D-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%7D%5Cend%7Barray%7D%5Cright%5D%2B%5Cleft%5B&#43;%5Cbegin%7Barray%7D%7Bl%7D%7B%5Cboldsymbol%7BB%7D%7D&#43;%5C%5C&#43;%7B0%7D%5Cend%7Barray%7D%5Cright%5D&#43;%5Cboldsymbol%7Bu%7D"
        title="[公式]" /></p>
<p>我们希望跟踪误差趋近于0并且其衰减速度可以调控，显然这取决于矩阵 <img
        class="lazyload"
        src="/svg/loading.min.svg"
        data-src="https://www.zhihu.com/equation?tex=A-LC"
        data-srcset="https://www.zhihu.com/equation?tex=A-LC, https://www.zhihu.com/equation?tex=A-LC 1.5x, https://www.zhihu.com/equation?tex=A-LC 2x"
        data-sizes="auto"
        alt="https://www.zhihu.com/equation?tex=A-LC"
        title="[公式]" /> 的特征根，这就是所谓的极点配置的问题：</p>
<ul>
<li>选择一个合适的 <img
        class="lazyload"
        src="/svg/loading.min.svg"
        data-src="https://www.zhihu.com/equation?tex=L"
        data-srcset="https://www.zhihu.com/equation?tex=L, https://www.zhihu.com/equation?tex=L 1.5x, https://www.zhihu.com/equation?tex=L 2x"
        data-sizes="auto"
        alt="https://www.zhihu.com/equation?tex=L"
        title="[公式]" /> ，使得 <img
        class="lazyload"
        src="/svg/loading.min.svg"
        data-src="https://www.zhihu.com/equation?tex=%5Coperatorname%7BRe%7D%5C%7B%5Clambda%28%5Cboldsymbol%7BA%7D-%5Cboldsymbol%7BL%7D&#43;%5Cboldsymbol%7BC%7D%29%5C%7D%3C0&#43;%5Cquad&#43;%5CLongrightarrow&#43;%5Cquad&#43;%5Clim&#43;_%7Bt&#43;%5Crightarrow&#43;%5Cinfty%7D%5Bx%28t%29-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%28t%29%5D%3D0"
        data-srcset="https://www.zhihu.com/equation?tex=%5Coperatorname%7BRe%7D%5C%7B%5Clambda%28%5Cboldsymbol%7BA%7D-%5Cboldsymbol%7BL%7D&#43;%5Cboldsymbol%7BC%7D%29%5C%7D%3C0&#43;%5Cquad&#43;%5CLongrightarrow&#43;%5Cquad&#43;%5Clim&#43;_%7Bt&#43;%5Crightarrow&#43;%5Cinfty%7D%5Bx%28t%29-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%28t%29%5D%3D0, https://www.zhihu.com/equation?tex=%5Coperatorname%7BRe%7D%5C%7B%5Clambda%28%5Cboldsymbol%7BA%7D-%5Cboldsymbol%7BL%7D&#43;%5Cboldsymbol%7BC%7D%29%5C%7D%3C0&#43;%5Cquad&#43;%5CLongrightarrow&#43;%5Cquad&#43;%5Clim&#43;_%7Bt&#43;%5Crightarrow&#43;%5Cinfty%7D%5Bx%28t%29-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%28t%29%5D%3D0 1.5x, https://www.zhihu.com/equation?tex=%5Coperatorname%7BRe%7D%5C%7B%5Clambda%28%5Cboldsymbol%7BA%7D-%5Cboldsymbol%7BL%7D&#43;%5Cboldsymbol%7BC%7D%29%5C%7D%3C0&#43;%5Cquad&#43;%5CLongrightarrow&#43;%5Cquad&#43;%5Clim&#43;_%7Bt&#43;%5Crightarrow&#43;%5Cinfty%7D%5Bx%28t%29-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%28t%29%5D%3D0 2x"
        data-sizes="auto"
        alt="https://www.zhihu.com/equation?tex=%5Coperatorname%7BRe%7D%5C%7B%5Clambda%28%5Cboldsymbol%7BA%7D-%5Cboldsymbol%7BL%7D&#43;%5Cboldsymbol%7BC%7D%29%5C%7D%3C0&#43;%5Cquad&#43;%5CLongrightarrow&#43;%5Cquad&#43;%5Clim&#43;_%7Bt&#43;%5Crightarrow&#43;%5Cinfty%7D%5Bx%28t%29-%5Chat%7B%5Cboldsymbol%7Bx%7D%7D%28t%29%5D%3D0"
        title="[公式]" /></li>
<li>如果 <img
        class="lazyload"
        src="/svg/loading.min.svg"
        data-src="https://www.zhihu.com/equation?tex=%5Coperatorname%7BRe%7D%5C%7B%5Clambda%28A-L&#43;C%29%5C%7D%3C-%5Csigma"
        data-srcset="https://www.zhihu.com/equation?tex=%5Coperatorname%7BRe%7D%5C%7B%5Clambda%28A-L&#43;C%29%5C%7D%3C-%5Csigma, https://www.zhihu.com/equation?tex=%5Coperatorname%7BRe%7D%5C%7B%5Clambda%28A-L&#43;C%29%5C%7D%3C-%5Csigma 1.5x, https://www.zhihu.com/equation?tex=%5Coperatorname%7BRe%7D%5C%7B%5Clambda%28A-L&#43;C%29%5C%7D%3C-%5Csigma 2x"
        data-sizes="auto"
        alt="https://www.zhihu.com/equation?tex=%5Coperatorname%7BRe%7D%5C%7B%5Clambda%28A-L&#43;C%29%5C%7D%3C-%5Csigma"
        title="[公式]" /> 就可以保证所有的初始值都可以按快于 <img
        class="lazyload"
        src="/svg/loading.min.svg"
        data-src="https://www.zhihu.com/equation?tex=%5Cmathrm%7Be%7D%5E%7B-%5Csigma&#43;t%7D"
        data-srcset="https://www.zhihu.com/equation?tex=%5Cmathrm%7Be%7D%5E%7B-%5Csigma&#43;t%7D, https://www.zhihu.com/equation?tex=%5Cmathrm%7Be%7D%5E%7B-%5Csigma&#43;t%7D 1.5x, https://www.zhihu.com/equation?tex=%5Cmathrm%7Be%7D%5E%7B-%5Csigma&#43;t%7D 2x"
        data-sizes="auto"
        alt="https://www.zhihu.com/equation?tex=%5Cmathrm%7Be%7D%5E%7B-%5Csigma&#43;t%7D"
        title="[公式]" /> 的速度收敛</li>
</ul>
<p>而根据对偶原理，对于矩阵 <img
        class="lazyload"
        src="/svg/loading.min.svg"
        data-src="https://www.zhihu.com/equation?tex=A-LC"
        data-srcset="https://www.zhihu.com/equation?tex=A-LC, https://www.zhihu.com/equation?tex=A-LC 1.5x, https://www.zhihu.com/equation?tex=A-LC 2x"
        data-sizes="auto"
        alt="https://www.zhihu.com/equation?tex=A-LC"
        title="[公式]" /> 的极点配置的问题相当于利用状态反馈 <img
        class="lazyload"
        src="/svg/loading.min.svg"
        data-src="https://www.zhihu.com/equation?tex=L%5ET"
        data-srcset="https://www.zhihu.com/equation?tex=L%5ET, https://www.zhihu.com/equation?tex=L%5ET 1.5x, https://www.zhihu.com/equation?tex=L%5ET 2x"
        data-sizes="auto"
        alt="https://www.zhihu.com/equation?tex=L%5ET"
        title="[公式]" /> 对于系统 <img
        class="lazyload"
        src="/svg/loading.min.svg"
        data-src="https://www.zhihu.com/equation?tex=%28A%5ET%2CC%5ET%29"
        data-srcset="https://www.zhihu.com/equation?tex=%28A%5ET%2CC%5ET%29, https://www.zhihu.com/equation?tex=%28A%5ET%2CC%5ET%29 1.5x, https://www.zhihu.com/equation?tex=%28A%5ET%2CC%5ET%29 2x"
        data-sizes="auto"
        alt="https://www.zhihu.com/equation?tex=%28A%5ET%2CC%5ET%29"
        title="[公式]" /> 的极点配置问题，而这等价于系统 <img
        class="lazyload"
        src="/svg/loading.min.svg"
        data-src="https://www.zhihu.com/equation?tex=%28A%5ET%2CC%5ET%29"
        data-srcset="https://www.zhihu.com/equation?tex=%28A%5ET%2CC%5ET%29, https://www.zhihu.com/equation?tex=%28A%5ET%2CC%5ET%29 1.5x, https://www.zhihu.com/equation?tex=%28A%5ET%2CC%5ET%29 2x"
        data-sizes="auto"
        alt="https://www.zhihu.com/equation?tex=%28A%5ET%2CC%5ET%29"
        title="[公式]" /> 可控，那么即等价于系统 <img
        class="lazyload"
        src="/svg/loading.min.svg"
        data-src="https://www.zhihu.com/equation?tex=%28A%2CC%29"
        data-srcset="https://www.zhihu.com/equation?tex=%28A%2CC%29, https://www.zhihu.com/equation?tex=%28A%2CC%29 1.5x, https://www.zhihu.com/equation?tex=%28A%2CC%29 2x"
        data-sizes="auto"
        alt="https://www.zhihu.com/equation?tex=%28A%2CC%29"
        title="[公式]" /> 可观。这里就解释了可观和可控的对偶的用处以及系统可观性的实际含义。</p>
<h2 id="参考">参考</h2>
<ol>
<li><a href="https://www.zhihu.com/people/feiyuxiaoTHU/answers/by_votes?page=2#ref_1_0" target="_blank" rel="noopener noreffer">^</a>段广仁. 线性系统理论. 哈尔滨工业大学出版社, 1996 <a href="https://item.jd.com/30834945675.html" target="_blank" rel="noopener noreffer">https://item.jd.com/30834945675.html</a></li>
<li><a href="https://www.zhihu.com/people/feiyuxiaoTHU/answers/by_votes?page=2#ref_2_0" target="_blank" rel="noopener noreffer">^</a>吴麒主编. 自动控制原理 (第2版, 下册). 清华大学出版社, 2006. <a href="http://www.tup.tsinghua.edu.cn/booksCenter/book_02284603.html" target="_blank" rel="noopener noreffer">http://www.tup.tsinghua.edu.cn/booksCenter/book_02284603.html</a></li>
<li><a href="https://www.zhihu.com/people/feiyuxiaoTHU/answers/by_votes?page=2#ref_3_0" target="_blank" rel="noopener noreffer">^</a>Discrete Control System, Katsuhiko Ogata  <a href="https://www.goodreads.com/book/show/241242.Discrete_Time_Control_Systems" target="_blank" rel="noopener noreffer">https://www.goodreads.com/book/show/241242.Discrete_Time_Control_Systems</a></li>
</ol>
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